Sure, let's simplify the given expression step-by-step:
[tex]\[
\left(-\frac{1}{2}\right)^2 \times \left(-\frac{1}{2}\right)^3
\][/tex]
1. Understand the expression: We are dealing with a product of two terms involving the same base [tex]\(\left(-\frac{1}{2}\right)\)[/tex] raised to different exponents, 2 and 3.
2. Apply the property of exponents: When multiplying terms with the same base, you can add the exponents:
[tex]\[
a^m \times a^n = a^{m+n}
\][/tex]
In this case, our base [tex]\(a\)[/tex] is [tex]\(-\frac{1}{2}\)[/tex], [tex]\(m=2\)[/tex], and [tex]\(n=3\)[/tex].
[tex]\[
\left(-\frac{1}{2}\right)^2 \times \left(-\frac{1}{2}\right)^3 = \left(-\frac{1}{2}\right)^{2+3}
\][/tex]
3. Add the exponents:
[tex]\[
2 + 3 = 5
\][/tex]
So the expression simplifies to:
[tex]\[
\left(-\frac{1}{2}\right)^5
\][/tex]
4. Evaluate the final expression: Calculate the power:
First, recognize that raising [tex]\(-\frac{1}{2}\)[/tex] to an odd power (5) will keep the negative sign.
[tex]\[
\left(-\frac{1}{2}\right)^5 = -\left(\frac{1}{2}\right)^5
\][/tex]
Next, calculate [tex]\(\left(\frac{1}{2}\right)^5\)[/tex]:
[tex]\[
\left(\frac{1}{2}\right)^5 = \frac{1^5}{2^5} = \frac{1}{32}
\][/tex]
Therefore,
[tex]\[
\left(-\frac{1}{2}\right)^5 = -\frac{1}{32}
\][/tex]
So, the simplified result of the expression [tex]\(\left(-\frac{1}{2}\right)^2 \times \left(-\frac{1}{2}\right)^3\)[/tex] is:
[tex]\[
-\frac{1}{32} \quad \text{or} \quad -0.03125
\][/tex]
Thus, the final answer is:
[tex]\[
-0.03125
\][/tex]
